Bulk and boundary S-matrices for the SU(N) chain

We consider both closed and open integrable antiferromagnetic chains constructed with the SU(N)-invariant R-matrix. For the closed chain, we extend the analyses of Sutherland and Kulish — Reshetikhin by considering also complex “string” solutions of the Bethe ansatz equations. Such solutions are essential to describe general multiparticle excited states. We also explicitly determine the SU(N) quantum numbers of the states. In particular, the model has particle-like excitations in the fundamental representations [k] of SU(N), with k = 1, …, N − 1. We directly compute the complete two-particle S-matrices for the cases [1] ⊗ [1] and [1] ⊗ [N − 1 ]. For the open chain with diagonal boundary fields, we show that the transfer matrix has the symmetry SU(l) × SU(N−l) × U(1), as well as a new “duality” symmetry which maps l→N−l. With the help of these symmetries, we compute by means of the Bethe ansatz for particles of types [1] and [N − 1 ] the corresponding boundary S-matrices.     

Seiberg-Witten geometry with various matter contents

We obtain the Seiberg-Witten geometry for four-dimensional N = 2 gauge theory with gauge group SO(2N_c) (N_c ⩽ 5) with massive spinor and vector hypermultiplets by considering the gauge symmetry breaking in the N = 2 E₆ theory with massive fundamental hypermultiplets. In a similar way the Seiberg-Witten geometry is determined for N = 2 SU(N_c) (N_c ⩽ 6) gauge theory with massive antisymmetric and fundamental hypermultiplets. Whenever possible we compare our results expressed in the form of ALE fibrations with those obtained by geometric engineering and brane dynamics, and find a remarkable agreement. We also show that these results are reproduced by using N = 1 confining phase superpotentials.     

Heavy quark and gluino potentials to two loops

The static potentials for systems of a heavy quark and anti-quark, two gluinos and a quark and a gluino are computed for all possible color configurations of a SU(Nc) gauge group.     

Quantum phase transitions in cascading gauge theory

We study N=1 supersymmetric SU(K+P)×SU(K) cascading gauge theory of Klebanov et al. (2000) [1] and [2] on R×S³ at zero temperature, and at the origin of the baryonic branch. A radius of S³ sets a compactification scale μ. An interplay between μ and the strong coupling scale Λ of the theory leads to an interesting pattern of quantum phases of the system. For μ?μχSB=1.240467(8)Λ the vacuum state of the theory is chirally symmetric. At μ=μχSB the theory undergoes the first-order transition to a phase with spontaneous breaking of the chiral symmetry. We further demonstrate that the chirally symmetric state of cascading gauge theory becomes perturbatively unstable at scales below μc=0.950634(5)μχSB. Finally, we point out that for μ<1.486402(5)Λ the stress-energy tensor of cascading gauge theory can source inflation of a closed Universe.     

THE MULTISYNCHRO-SPHERICAL SOLUTIONS OF THE SU(N)GAUGE THEORY

We study the monopole solutions of the SU(N) gauge group systematically,generalize the current synchro-spherical symmetrical monopole to the multi synchro-spherical symmetrical case,and give the complete classification of the monopole so-lutions.As examples,we have calculated the SU(3) and SU(4) monopoles in detail,and discuss some problems such as angular momenta and topological quantum num-bers.     

The nonlocal chiral quark model and the muon <Emphasis Type="Italic">g</Emphasis> – 2 problem

In the first part of the review we discuss the effective nonlocal approach in the quantum field theory. It concerns primary the historical retrospective of this approach, and than we concentrate on the interaction of matter particles (fermions and bosons) with the (abelian and nonabelian) gauge fields. In the second part of the review we consider the hadronic corrections (vacuum polarization) to the anomalous magnetic moment of the muon g - 2 factor discussed within the SU_f(2) nonlocal chiral quark model. This is considered in the leading and, partially, in the next-to-leading orders (the effect of the fermion propagator dressing due to pion field) of expansion in small parameter 1/N_c (N_c is the number of colors in QCD).     

Noncommutative SU(N) and gauge invariant baryon operator

We propose a constraint on the noncommutative gauge theory with U(N) gauge group which gives rise to a noncommutative version of the SU(N) gauge group. The baryon operator is also constructed.     

Guided random walks for solving Hamiltonian lattice gauge theories

Motivated by developments for many-particle quantum systems, a Monte Carlo method for solving Hamiltonian lattice gauge theories without fermions is presented in which a stochastic random walk is guided by a trial wave function. To the extent that a substantial portion of the local structure of the theory can be incorporated in the trial function, the method offers significant advantages relative to existing techniques. The method is applicable to the study of SU(N) lattice gauge theories, and its utility is demonstrated by solving the compact U(1) gauge theory in three spatial dimensions.     

Gluon-ghost condensate of mass dimension 2 in the Curci-Ferrari gauge

The effective potential for an on-shell BRST invariant gluon-ghost condensate of mass dimension 2 in the Curci-Ferrari gauge in SU(N) Yang-Mills is analysed by combining the local composite operator technique with the algebraic renormalization. We pay attention to the gauge parameter independence of the vacuum energy obtained in the considered framework and discuss the Landau gauge as an interesting special case.     

The trinification model <Emphasis Type="Italic">SU</Emphasis>(3)<Superscript>3</Superscript> from orbifolds for fuzzy spheres

In this review, we consider an N = 4 supersymmetric SU(3N) gauge theory defined on the Minkowski spacetime. Then we apply an orbifold projection leading to an N = 1 supersymmetric SU(N)³ model, with a truncated particle spectrum. Then, we present the dynamical generation of (twisted) fuzzy spheres as vacuum solutions of the projected field theory, breaking the SU(N)³ spontaneously to a chiral effective theory with unbroken gauge group the trinification group, SU(3)³.     

Discrete symmetries in SO(N) grand unified groups and the survival hypothesis

The possibility of introducing the discrete symmetry D is studied, which along with SU^c(3) × SU_L(2) × U(1)-symmetry, remains after the breaking of SO(N) grand unified symmetry by the Higgs fields vevs ～ 10¹⁵ GeV. The D quantum number distinguishes the fermions coupled with W-bosons via left and right currents. As a result, the presence of low-mass fermions in the theory is provided.     

The QCD Hamiltonian and nonlinear quantum mechanics

We study the canonical quantization of SU(N) gauge theory in linear, noncovariant gauges. The canonical formalism is first discussed for the classical theory, with special attention to the features involving nonlinearity and the gauge degrees of freedom. The transition to the quantum theory is then performed for an arbitrary linear gauge, using the covariant quantization rules of nonlinear quantum mechanics. When the quantum Hamiltonian is written in the Weyl-ordered form appropriate for the application of the usual Dyson-Wick perturbative techniques, additional ordering terms appear with respects to the classical Hamiltonian. We discuss the relation of our results to those of previous authors, and the relevance of the ordering terms in field theory.     

Polynomial invariants for SU(2) monopoles

We present an explicit expression for the topological invariants associated to SU(2) monopoles in the fundamental representation on spin four-manifolds. The computation of these invariants is based on the analysis of their corresponding topological quantum field theory, and it turns out that they can be expressed in terms of Seiberg-Witten invariants. In this analysis we use recent exact results on the moduli space of vacua of the untwisted N = 1 and N = 2 supersymmetric counterparts of the topological quantum field theory under consideration, as well as on electric-magnetic duality for N = 2 supersymmetric gauge theories.     

Semi-classical spectrum of the homogeneous sine-Gordon theories

The semi-classical spectrum of the homogeneous sine-Gordon theories associated with an arbitrary compact simple Lie group G is obtained and shown to be entirely given by solitons. These theories describe quantum integrable massive perturbations of Gepner's G-parafermions whose classical equations-of-motion are non-abelian affine Toda equations. One-soliton solutions are constructed by embeddings of the SU(2) complex sine-Gordon soliton in the regular SU(2) subgroups of G. The resulting spectrum exhibits both stable and unstable particles, which is a peculiar feature shared with the spectrum of monopoles and dyons in N = 2 and N = 4 supersymmetric gauge theories.     

Chiral rings,singularity theory and electric-magnetic duality

We study in detail the space of perturbations of a pair of dual N = 1 supersymmetric theories based on an SU(N_c) gauge theory with an adjoint X and fundamentals with a superpotential which is polynomial in X. The equivalence between them depends on non-trivial facts about polynomial equations, i.e. singularity theory. The classical chiral rings of the two theories are different. Quantum mechanically there are new relations in the chiral rings which ensure their equivalence. Duality interchanges “trivial” classical relations in one theory with quantum relations in the other and vice versa. We also speculate about the behavior of the theory without the superpotential.     

Low mass-scale parity restoration in expanded gauge theories

It is shown that schemes of grand unification withSU(2n)⁴ gauge symmetry permit the embedding of the left-right symmetricSU(2) _L ×SU(2) _R ×U(1)×SU(3) ^c intermediate symmetry at relativelylow energies [between 250 GeV and a few TeV] as well as allows light unification mass-scales (≦10⁵ TeV) ifn≧3 for values of the weak angle Sin² θ _W and the strong coupling α _S in the ranges 0.20≦Sin² θ _W ≦0.25, 0.10≦α _S ≦0.15.     

A low composite scale preon model with complementarity

We have constructed the first “realistic candidate” preon model with low composite scale satisfying complementarity between the Higgs and confining phases. The model is based onSU(4) metacolor and predicts four generations of ordinary quarks and leptons together with heavy neutrinos at the level of the standard gauge groupSU(3) _c ×SU(2) _L ×U(1) _Y . There are no exotic massless fermions. The global family group isSU(2)×U(1).     

Dirac monopoles embedded in <Emphasis Type="Italic">SU</Emphasis>(<Emphasis Type="Italic">N</Emphasis>) gauge theory with the θ term

Dirac monopoles embedded in SU(N) gauge theory with the θ term are considered. For θ=4πM (where M is half-integer and integer for N=2 and N>2, respectively), these monopoles acquire an SU(N) charge due to the θ term and become dyons. They belong to various (but not any) irreducible representations of the SU(N) group. The admissible representations are listed. Their minimum dimension increases with N. The basic result of the study is the representation of the partition function of any SU(N) model involving the θ term and complemented by singular gauge fields corresponding to the indicated monopoles in the form of a vacuum average of the product of Wilson loops viewed along the world lines of the monopoles. This vacuum average must be calculated in the corresponding model without the θ term.     

The large-N limit in statistical physics and the one-particle Schrödinger equation

Nonlinear d-dimensional vector σ models, such as O(N), SU(N), and CP ^N, are considered in the limit of an infinite number of components N. It is shown that the equation for the two-point correlation function in these models is similar to the Schrödinger equation for a quantum particle moving in a δ-function potential well (?T)δ(x), where T is the temperature. This equation adequately describes the systems under study both above and below the Curie point. Within this approach, the critical behavior of the SU(N)-invariant Ginzburg-Landau model in an external uniform magnetic field is determined in the vicinity of the upper critical magnetic field. The critical indices in this case are the same as in the spherical model in a random magnetic field. An exact equation describing the H _c2(T) curve of continuous phase transitions is derived, which allows one to determine the asymptotes of this curve in strong and weak fields. The relation between the one-particle Schrödinger equation and critical phenomena is analyzed, and applications of this method to various models in solid state physics and statistical mechanics are discussed.     

Color Superconductivity in Supersymmetric Gauge Theories

The studies of superconductivity, dual superconductivity and color superconductivity have been undertaken through the breaking of supersymmetric gauge theories which automatically incorporate the condensation of monopoles and dyons leading to confining and superconducting phases. Constructing the total effective Lagrangian of N=2 SU(2) gauge theory with N _f =2 quark multiplets and quark chemical potential at classical and quantum levels, it has been demonstrated that baryon number symmetry is spontaneously broken as a consequence of the SU(2) strong gauge dynamics and the color superconductivity dynamically takes space at the non-SUSY vacuum.